What does Atticus say is the one thing that “doesn’t abide by majority rule”?

The answer to this is that a person's conscience is the one thing that, according to Atticus, does not abide by majority rule.  He says that he has to be able to live with himself and therefore he has to follow his own conscience.

You can find this in Chapter 11.  Scout tells Atticus he must be wrong about defending Tom Robinson.  She says this because so many other people say Atticus is wrong.  At that point, Atticus says he could not live with himself if he did not do what he thought was right.